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Exceptional Circle Helsinki Workshop 2013

This is a five-day scientific workshop held at the Department of Mathematics and Statistics of the University of Helsinki on August 12-16, 2013.

The topics of the workshop include nonlinear evolution equations, random matrix theory and inverse scattering methods. The workshop is a follow-up meeting to the Focussed Research Group meeting The d-bar method: Inverse scattering, nonlinear waves, and random matrices held in Banff International Research Station in July-August 2012.

Special emphasis is placed on the analysis and numerical solution of the Novikov-Veselov (NV), Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations using the so-called D-bar methodology based on inverse scattering.

The name Exceptional Circle arises from the typical singularities arising in the scattering data under suitable circumstances. See the related article.

 


Invited speakers

Ken McLaughlin (University of Arizona, USA) 
Anna Kazeykina (CMAP, Ecole Polytechnique
Peter Miller (University of Michigan, USA)
Michael Music (University of Kentucky, USA)
Peter Perry (University of Kentucky, USA) 
Andreas Stahel (Berner Fachhochschule Technik und Informatik, Switzerland



Programme

The talks will be in seminar room B322 in the Exactum building.

Monday, August 12
09:30 Coffee (in 4th floor coffee room)
10:00 Opening 
10:10 Peter Perry: Determinants and Inverse Scattering
11:00 Anna Kazeykina: Novikov-Veselov Equation and Solitons 
11:30 Lunch
13:00 Peter Miller: Semiclassical Analysis for Linear and Nonlinear Waves
13:30 Ken McLaughlinSome questions in the asymptotic analysis of integrable systems in 2+1 dimensions
14:00 Michael Music: Inverse Scattering for Subcritical Potentials  
14:30 Coffee (in 4th floor coffee room), planning the rest of the week

Tuesday, August 13: Computational day
09:00 Samuli Siltanen: Numerical implementation of the inverse scattering method
09:30 Sarah Hamilton: Computational results for the Davey-Stewartson equation 
10:00 Coffee (in 4th floor coffee room)
10:30 Andreas Stahel: Direct numerical solution of nonlinear evolution equations
11:00 Janne Tamminen: Positive-energy exceptional points
11:30 Discussion
12:00 Lunch
13:30 Computational workshop / Research discussions 

Wednesday, August 14

 

Morning: Research discussions 

13:00 Anna Kazeykina: Non-zero energy scattering

Thursday, August 15

Morning: Research discussions 

19:00 Conference dinner at restaurant Lappi, located in the center of Helsinki in the address Annankatu 22.

Friday, August 16:

Excursion to Haltia Nature Center.

9:30 The bus leaves near the Kiasma museum main entrance. The round-trip costs 20 euros for adults and 12 euros for kids, and includes the exhibition entrance fee. CASH ONLY! This is the route from the guesthouse to the place where the bus leaves from:

The place where the bus leaves is marked with an arrow here:

Once we get to Haltia, there will be exhibition viewing, lunch, and hiking. Please bring sneakers or other shoes suitable for hiking in the forest. And, just in case, raingear.

We can pick berries, such as lingonberries:

15:00 The bus leaves from Haltia center back to Helsinki.


Abstracts:


Michael Music: Inverse Scattering for Subcritical Potentials  

In a 1996 paper, Nachman worked out properties of the scattering transform for the Schrödinger equation in the case where the potential is conductivity-type. We show that many of these properties can be extended to another class of potentials called "subcritical" by Murata. As with conductivity-type, these potentials have a positive solution to the corresponding Schrödinger equation, but this solution is not bounded. In particular, for subcritical potentials we show that the scattering transform has no exceptional points, but its behavior near zero is worse than that for the scattering transforms of conductivity-type potentials. 

 

Peter Perry: Determinants and Inverse Scattering

A fundamental problem in inverse scattering for dispersive equations in two dimensions is to characterize the exceptional set of points for which CGO solutions are not uniquely defined. At such points, a Fredholm operator associated to the defining equation for CGO solutions has a nontrivial kernel. Here we'll show how to use work of Gohberg, Goldberg, and Krupnik to define and analyze an associated Fredholm determinant whose zeros are the points of the exceptional set, and  illuminate the relationship between singularities of the scattering transform and the existence of exceptional points. This work is joint with Michael Music.


Samuli Siltanen: Numerical implementation of the inverse scattering method
The inverse scattering method is based on the use of so-called complex geometric optics (CGO) solutions that behave asymptotically as the exponential function. CGO solutions can be used to construct adaptive non-linear Fourier transforms that linearize certain nonlinear evolution equations. Such equations include Korteveg-de Vries, Davey-Stewartson, Kadomtsev-Petviashvili and Novikov-Veselov equations. These equations can be solved numerically by implementing appropriate direct and inverse nonlinear Fourier transformations. This in turn can be done using a variant of Gennadi Vainikko's solution method for the Lippmann-Schwinger equation, based on periodization and FFT-friendly discretization.


Andreas Stahel: Direct numerical solution of nonlinear evolution equations
The Novikov-Veselov equations are a system of PDE's, leading to a nonlinear evolution equation. We present a Fourier based approach to find numerical approximation to the solutions of the evolution system. This spectral method is at least competitive with standard finite difference approximations. We give some stability results and apply the method to a few sample situations.



Travel:

To/From Helsinki-Vantaa Airport:  From outside baggage claim (in both Terminals 1&2) you will find taxis.  A taxi from the airport to Töölö Towers (Pohjoinen Hesperiankatu 23) costs around 40€, can be paid with cash or credit card, and will be reimbursed by the conference.  When returning to the airport, the staff at the Towers can schedule a taxi to pick you up.

 

To/From the University-Töölö Towers:  Public transportation is the easiest way to travel from the Towers to Kumpula Campus where the conference will take place.  Walk southeast Exiting Töölö Towers, take a left and walk south across the park towards the SIWA (a small grocery store) and tram stop).  Directly across from SIWA (on Runeberginkatu) you can pick up Tram 8 heading northeast (cross the tracks) or Bus 70T (also across the tracks).

  • Take Tram 8 from Töölö (tram stop Apollonkatu) to Kumpula 
  • Take Bus 70T from Töölö (bus stop Apollonkatu) to Kumpula (bus stop Kumpulan Kampus).

The mathematics building, called Exactum, is across the street on top of the hill. See the campus map.

You can buy a ticket on board (using cash) for 2.80€ (good for 1 hour 15 minutes).  It may be convenient to have a 5 day pass.  You can buy a pass from the R-Kioski (the Finnish 7/11) near the bus stop.  Ask the shop owner (English is fine) for a ____ day pass (see HSL website for fares).  The pass is good for unlimited travel by tram, bus, and the Suomenlinnna ferry during its period of validity.  The passes are typically not valid during 2-4am when fares are more expensive.  The Journey Planner website determines your travel options between locations for a given time:

 

Via Tram 8

 

 

Via Bus 70T

 


Dining Options:

Near Töölö Towers you have various dining options.  I suggest checking out TripAdvisor for personalized recommendations, there is also a free TripAdvisor downloadable city guide app for iPhone (and probably android) that works offline, just download the one for Helsinki.  Furthermore, your apartments have efficiency kitchens.  You can buy groceries from the SIWA across the park, the Alepa (past the SIWA at the Y-intersection) or at a number of other grocery stores.  At the reception office in Töölö Towers you can find city maps and a local map with restaurants/shops for your immediate area.  If you have further questions, just ask Sarah or Samuli.

 


What to do in Helsinki

Some ideas about how to spend your spare time in the Helsinki area are collected to this page.

 


Organizers:

Sarah Hamilton and Samuli Siltanen.

 

 

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